Hydrodynamics of slender swimmers near deformable interfaces

We study the coupled hydrodynamics between a motile slender microswimmer and a deformable interface that separates two Newtonian fluid regions. From the disturbance field generated by the swimming motion, we quantitatively characterize the interface deformation and the manner in which the coupling modifies the microswimmer translation itself. We treat the role of the swimmer type (pushers and pullers), size and model an interface that can deform due to both surface tension and bending elasticity. Our analysis reveals a strong dependence of the hydrodynamics on the swimmer orientation and position. Given the viscosities of the two fluid media, the interface properties and the swimmer type, a swimmer can either migrate toward or away from the interface depending on its configurations. When the swimmer is oriented parallel to the interface, a pusher-type swimmer is repelled from the interface at short times if it is swimming in the more viscous fluid. At long times, however, pushers are always attracted to the interface, and pullers are always repelled from it. However, swimmers oriented orthogonal to the interface exhibit a migration pattern opposite to the parallel swimmers. In consequence, a host of complex migration trajectories emerge for swimmers arbitrarily oriented to the interface. We find that confining a swimmer between a rigid boundary and a deformable interface results in regimes of attraction toward both surfaces depending on the swimmer location in the channel, irrespective viscosity ratio. The differing migration patterns are most prominent in a region of order the swimmer size from the interface, where the slender swimmer model yields a better approximation to the coupled hydrodynamics.

Figure 1

A schematic representation of a slender swimmer translating near a deformable interface with speed $V_s$ along its director vector $\mathit{p}$. The swimmer is characterized by an aspect ratio $\kappa$ defined as the ratio of its total length $L$ to its lateral extent $W$. The circles along the axial length of the fore-aft symmetric swimmer represent a line distribution of stokeslets characterizing the head and tail for pushers. The disturbance flow field generated by the swimming motion deforms the interface, and $u_z$ is the interface deformation (solid gray line) relative to its initially flat undeformed (dotted line) state $z_0$.

Figure 2

The nondimensional interface deformation plotted for a swimmer of unit length oriented along the $r_x$ direction (a) viewed at an angle of ${30}^{\circ }$ from the undeformed interface and (b) top view. The black line in (b) spanning from $r_x\in [-1/2,1/2]$ signifies the slender swimmer. The colorbar given on the right-hand side of (b) applies to both figures. The plot is for $\mathrm{\Gamma }=0.1$, $\lambda =0.5$ at $t\approx 2$, $\kappa =10,$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 3

The interface deformation, $u_z$, plotted for a swimmer oriented along the $r_x-\mathrm{coordinate}$ for three values of $\mathrm{\Gamma }=$ 0.1, 1 and 5 (a) along $r_x$ (b) along $r_y$. In both the figures $\lambda =0.5$, $r_{z_0}\left(0\right)=1$, $t\approx 2$, $\kappa =10,$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$. Legends under $\circ$ refer to $u_z$ obtained from solving the pair of Eqs. (12) and (13), and those under $\diamond$ from solving the approximate Eqs. (15) and (16). The (black) dotted line in the insets is the far-field approximation from Eq. (14).

Figure 4

The vertical swimmer translation velocity $V_z^T$ of pushers plotted as a function of the initial distance of the swimmer from the interface $r_{z_0}\left(0\right)$ at different time instants for the viscosity ratios (a) $\lambda =0.5$, (b) $\lambda =1,$ and (c) $\lambda =1.5$. The plot is to be interpreted as follows: at each $r_{z_0}\left(0\right)$, the vertical black dotted lines trace the time evolution of $V_z^T$. In all the plots $\kappa =10$, $\mathrm{\Gamma }=1,$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 5

The relative vertical swimmer trajectory $r_{z_0}\left(t\right)-r_{z_0}\left(0\right)$ of pushers plotted as a function of time, for the viscosity ratios $\lambda =0.5$, 1 and 1.5. In all the plots $\kappa =10$, $\mathrm{\Gamma }=1$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 6

The relative swimmer trajectory $r_{z_0}\left(t\right)-r_{z_0}\left(0\right)$ of a pusher plotted as a function of time. In panel (a) the initial swimmer distance to the interface is fixed: $|r_{z_0}|=1$ and the ratio of surface tension to bending stress is varied: $\mathrm{\Gamma }=$ 0.1, 1, 10. In panel (b) both $r_{z_0}\left(0\right)$ and $\mathrm{\Gamma }$ are varied so as to yield similar swimmer translation, with $r_{z_0}\left(0\right)=1.5,2$ and $\mathrm{\Gamma }=1,4$, respectively. In both figures $\lambda =$ 0.5, $\kappa =10,$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$. In the inset the corresponding swimmer translation velocities $V_z^T$ are plotted for the same parameters.

Figure 7

The interface deformation, $u_z$, due to a pusher plotted as a function of the radial distance along the interface $r_{\parallel }$ for $\mathrm{\Gamma }=$ 0.1, 1, and 10 at time $t=2$ (a) for $r_{z_0}\left(0\right)=1$ and (b) $r_{z_0}\left(0\right)=5$. The dotted lines in the inset represent the far-field $O{\left(r_{\parallel }\right)}^{-3}$ scaling. Legends under $\circ$ refer to $u_z$ obtained from solving the pair of Eqs. (17) and (18) and those under $\diamond$ from solving Eqs. (20) and (21). The “Far-field” in the inset of (b) refers to $u_z$ from Eq. (19) for $\mathrm{\Gamma }=1$.

Figure 8

The vertical swimmer translation velocity $V_z^T$ of pushers plotted as a function of the initial distance of the swimmer from the interface $r_{z_0}\left(0\right)$ at different time instants for the viscosity ratios (a) $\lambda =0.5$, (b) $\lambda =1,$ and (c) $\lambda =1.5$. The plot is to be interpreted as follows: at each $r_{z_0}\left(0\right)$, the vertical black dotted lines trace the time evolution of $V_z^T$ as indicated by the black arrow. In all the plots $\kappa =10$, $\mathrm{\Gamma }=1$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 9

The relative vertical swimmer trajectory $r_{z_0}\left(t\right)-r_{z_0}\left(0\right)$ of pushers plotted as a function of time, for the viscosity ratios $\lambda =$ 0.5, 1, and 1.5. In all the plots $\kappa =10$, $\mathrm{\Gamma }=1,$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 10

(a) The rotation rate ${\dot{p}}_z$ plotted as a function of $p_z(\equiv cos\theta )$. (b) The vertical component of the translation velocity $V_z^T\left(t\right)$ of pushers plotted as a function of time, for a range of initial swimmer orientation relative to the $r_z$ axis $\theta \left(0\right)={30}^{\circ }$, ${45}^{\circ }$, ${60}^{\circ }$, ${75}^{\circ }$, ${85}^{\circ }$, and $89.5^{\circ }$. In the inset, $V_z^T$ is shifted by the time it takes the $\theta \left(0\right)=89.5^{\circ }$ curve to attain the corresponding $\theta \left(0\right)$. In all the plots $\lambda =$ 0.5, $\kappa =10$, $\mathrm{\Gamma }=1$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 11

(a) The relative vertical swimmer trajectory $r_{z_0}\left(t\right)-r_{z_0}\left(0\right)$ of pushers plotted as a function of time for a range of initial swimmer orientation relative to the $r_z$ axis $\theta \left(0\right)={30}^{\circ }$, ${45}^{\circ }$, ${60}^{\circ }$, ${75}^{\circ }$, ${85}^{\circ }$ and $89.5^{\circ }$. The viscosity ratio is $\lambda =0.5$, and the inset is a zoomed version of the abscissa for $t\in [0,1]$. (b) The relative vertical swimmer trajectory $r_{z_0}\left(t\right)-r_{z_0}\left(0\right)$ of pushers plotted as a function of time, for the viscosity ratios $\lambda =$ 0.5, 1 and 1.5. In the inset the corresponding vertical swimmer translation velocity $V_z^T$ of pushers plotted as a function of time. In all the plots $\kappa =10$, $\mathrm{\Gamma }=1$, and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 12

The relative vertical swimmer trajectory $r_{z_0}\left(t\right)-r_{z_0}\left(0\right)$ of pushers and pullers plotted as a function of time, initial orientations $\theta \left(0\right)={30}^{\circ }$ and ${60}^{\circ }$. In the insets, the corresponding $V_z^T\left(t\right)$ plotted as a function of time. In all the plots $\lambda =0.5$, $\kappa =10$, $\mathrm{\Gamma }=1$ and ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=1$.

Figure 13

A sectional schematic representation of a slender swimmer translating with speed $V_s$ along its director vector $\mathit{p}$ in a channel with a deformable interface below and a rigid boundary on top. The distance between the plane of the undeformed interface and the rigid boundary is $H$. The circles along the axial length of the fore-aft symmetric swimmer represent a line distribution of stokeslets characterizing the head and tail for pushers. The disturbance flow field generated by the swimming motion deforms the interface, and $u_z$ is the interface deformation (solid gray line) relative to its initially flat undeformed (dotted line) state $z_0$.

Figure 14

(a) The relative swimmer trajectory $r_{z_0}\left(t\right)-r_{z_0}\left(0\right)$ of a pusher plotted as a function of time for different distance $H$ between the interface and the rigid boundary: $H=$ 2, 2.2, 2.4, 10, when the viscosity ratio is $\lambda =0.5$. (b) $r_{z_0}\left(t\right)-r_{z_0}\left(0\right)$ of a pusher plotted as a function of time for $\lambda =$ 0.5, 1, and 1.5, and fixing $H$ = 2.3. The insets contains the corresponding vertical component of the translation velocity $V_z^T\equiv d{r}_{z_0}/dt$. The “repulsion” and “attraction” are to be interpreted relative to the interface. In both figures, the initial swimmer distance to the interface is $|r_{z_0}|=1$ and the ratio of surface tension to bending stress is $\mathrm{\Gamma }=$ 1. The attraction and repulsion are specified relative to the deformable interface.

Figure 15

The interface deformation $u_z$ due to a pusher plotted as a function of the distance parallel to the interface $r_{\parallel }$ on a log-log scale for the ratio of viscous stress to bending stress: ${\eta }_1{V}_s{L}^2/{\kappa }_{\beta }=$ 1 and 10, at $t=1$ when $\lambda =0.5$ and $r_{z_0}\left(0\right)=1$. Solid lines represent the full solution obtained from Eq. (C6), and the dashed lines the approximate solution from Eq. (17). The inset shows a linear plot of $u_z$ for $\eta V_s{L}^2/{\kappa }_{\beta }=10$.